If you had any number of ordinary dice, what are the possible ways
of making their totals 6? What would the product of the dice be
each time?

Which Symbol?

Stage: 2 Challenge Level:

This problem challenged you to check
your understanding of the building blocks of maths: "plus",
"minus", "divide", and "multiply". You noticed that theorderof numbers and symbols is important; in
different positions, the "number sentence" can say different
things. This is the same as when you speak or write; the order of
the words matters. However, with some of the number sentences,
there are two different orders, which still mean the same thing.
Again, this is like speaking or writing; even if the order is
different, it can sometimes mean the same thing.

For the first four questions, you were
asked to put the correct symbol into the box. Several students
submitted correct solutions. These include: Jonathan, Jordan, and
Callum from Aycliffe Drive Primary School, Lauren from Princess
Elizabeth, Anna, Isabella, George, Sophie, and Rhiannon from St.
Swithun's, Nathan from Wilson's, Rebecca from Bourne Westfield
primary school, Ayush from Garden Gate Elementary School, Brandon,
Narissa, Jordan, Justin, and Cameron from Village Elementary, and
Charlotte from Manor Preparatory School.

Narissa and Jordan wrote out the answer:

$16+18=34$
$47-28=19$
$18\div 2=9$
$30=10\times 3$

Rhiannon, from St Swithun's Primary
School approached this problem by trial and improvement: she placed
a different symbol in the box, and looked to see if the calculation
made sense. In this way, she worked out the correct
symbols.

The next part
of the problem asked you to fill in the symbols, like in the first
section. However, all of the number sentences but two have two
different solutions. Ayush, from Garden Gate Elementary School
submitted the correct solution:

$51- 36 = 15$
$51 = 36 + 15$

$45\div 5 = 9$
$45 = 5\times 9$

$27 + 36 = 63$

$70-14 = 56$
$70 = 14 + 56$

$7\times 5 = 35$

$50\div 5 = 10$
$50 = 5\times 10$

As Aimee, from Culford School points out:

The two sentences that only have one solution are:

$7\times 5=35$ and
$30=10\times3$

Rebecca, from Bourne Westfield Primary
School explained:

On the $27+36=63$, the largest number is at the end so if you tried
to make a second number sentence like $27=36-63$ it wouldn't give
the right answer, because if you take away $63-36$ you would go
into negative numbers. So, to get two "working" number sentences,
you have to have the largest number at the front .
Here is an example to show a number sentence that works:
$50\div 5=10$ and $50=5\times 10$ .
This is a an example to show one that you can only do in one way:
$7\times 5=35$.

Charlotte, from Manor Preparatory School, also submitted the
correct solution.

A few people noticed a reason why there can be
two different solutions for the number sentence. As Jonathan,
Jordan and Callum, from Aycliffe Drive Primary School point out,
"plus" and "minus" are inverse operations (processes), as are
"multiply" and "divide". This means that you can do a sum, for
example a multiplication, and then undo it, by doing the reverse,
or "inverse": divide.

The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the
NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to
embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities
can be found here.